The motion of a dynamical system may be approximated as a sequence of discrete steps in time described by transfer maps. In the field of accelerator physics, Taylor series maps constitute a special, heavily-used class of such maps, which, despite their wide use, have poorly understood, or little appreciated, convergence properties. In Part I we show first how one may expect a (very general) transfer map to be analytic within some, perhaps quite limited, region of phase space. We then show that the underlying singularity structure of the original map--as determined by the dynamical system itself--governs the domain of convergence of a given Taylor series map. We conclude Part I by using the quartic anharmonic oscillator as an example to illustrate not only the complicated, rich, and very subtle behavior of such domains of convergence, but also the care and understanding required when drawing conclusions about the applicability of Taylor maps. Following a Hamiltonian flow for a finite interval of time produces a simplistic map. In Part II we describe a procedure for converting a truncated Taylor series approximation for a simplistic map into a polynomial map that is exactly simplistic--i.e., a Cremona map-- in such a way that the Cremona map agrees with the original Taylor map through terms of any desired order. We then introduce the concept of sensitivity vectors and show how that concept allows one to characterize optimal Cremona symplectifications. We also give explicit constructions for optimal Cremona symplectifications in two- and four- and six-dimensional phase spaces. At the end, we apply these methods to some maps of physical interest. We expect that Cremona maps may be useful for studying the long-term behavior of particles circulating in storage rings.